Let us point out that both the algorithms in
[1]} and [2]} have \(O(1/\sqrt{K})\)  expected convergence. However, the algorithm in [1]} is an extension of Zinkevich's online algorithm ([4]}), which is an variant of projection gradient method, and the CSA method in [2]} is a stochastic counterpart of Polyak's subgradient method [6]}. When problem (REF ) reduces to deterministic problem, these algorithms have at most linear rate of convergence. In contrast, SLPMM becomes the classical proximal method of multipliers, which has asymptotic superlinear rate of convergence.
Moreover, in [2]}, the iteration complexity analysis is based on the selection of stepsizes \(\lbrace \gamma _k\rbrace \)  and \(\lbrace \eta _k\rbrace \) , which are dependent on the parameters \(D_X, M_F\)  and \(M_G\) . However, these data are not known beforehand when problem (REF ) is put forward to solve. Note that, in SLPMM the stepsizes \(\sigma \)  and \(\alpha \)  are problem-independent.
